Expectation Values of Observables in Time-Dependent Quantum Mechanics

Abstract

Let U(t) be the evolution operator of the Schrödinger equation generated by a Hamiltonian of the form H ₀(t) + W(t), where H ₀(t) commutes for all twith a complete set of time-independent projectors $\{ P_j \} _{j = 1}^\infty$. Consider the observable A=∑_j P _jλ_jwhere λ _j ≃ j ^μ, μ>0, for jlarge. Assuming that the “matrix elements” of W(t) behave as EquationSource % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaauWaaeaaca% WGqbWaaSbaaSqaaiaadQgaaeqaaOGaam4vaiaacIcacaWG0bGaaiyk% aiaadcfadaWgaaWcbaGaam4AaaqabaaakiaawMa7caGLkWoacqWIdj% YocaaIXaGaai4lamaaemaabaGaamOAaiabgkHiTiaadUgaaiaawEa7% caGLiWoadaahaaWcbeqaaiaadchaaaGccaGGSaGaamOAaiabgcMi5k% aadUgaaaa!4E46! for p>0 large enough, we prove estimates on the expectation value $\langle U(t)\phi|AU(t)\phi\rangle\equiv\langle A\rangle_\phi(t)$for large times of the type EquationSource % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyykJeUaam% yqaiabgQYiXpaaBaaaleaaruqtV52B0LhCLbYqVj3CPzxyaGqbaiaa% -z8aaeqaaOGaaiikaiaadshacaGGPaGaeyizImQaam4yaiaadshada% ahaaWcbeqaaiabes7aKbaaaaa!49E5! where δ>0 depends on pand μ. Typical applications concern the energy expectation 〈H₀〉_ϕ(t) in case H ₀(t) ≡ H ₀or the expectation of the position operator 〈x²〉_ϕ(t) on the lattice where W(t) is the discrete Laplacian or a variant of it and H ₀(t) is a time-dependent multiplicative potential.